When not referring to a person, the white rook on the chessboard would be correct. When saying, White moved his rook to e1, or White's rook is now on e1, then the word white is capitalized.
The words White and Black are capitalized when in reference to a person.
Examples:
The move 1...c5 introduces the Symmetrical English, which some think is Black's safest way to counter 1) c4.
See how White has ripped away Black's protective pawn-cover?
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Pieces are not capitalized. It would be harder to read chess books if both the words White and Black AND all of the pieces are capitalized.
Examples:
White has the simple idea of Bb2 and later moving his knight from c3 to exchange the important g7-bishop; then he may be able to exploit Black's resulting dark-square vulnerabilities on the kingside.
In large part this is because Black hasn't committed his king's knight.
Still better, Black can play 2...Bc5(!) right away, to keep open the idea of ...c6, which would repel a knight from d5.
How do we capitalize things in chess? Some people capitalize all the piece names and sides (like White Rook, White's Rook), or the piece names (white Rook), while others like me prefer not to capitalize them (white rook).